Is it sufficient to wash my niece's clothes, which have become impure, once in a single vessel, and what is the ruling on doubting the purity of the house due to her repeated acts of impurity and her mother's lack of concern?

All praise is due to Allah, and prayers and peace be upon the Messenger of Allah. And thereafter: Beware of abandoning prayer, for it is one of the gravest sins. A clean mat or prayer rug can be used for prayer. The default state of things is purity, so do not deem something impure merely based on doubt. As for clothes that have become impure, it is better to wash them outside the container before placing them in it. Beware of obsessive whispering (waswasah) and completely turn away from it. And Allah knows best.